Introduction to Dirichlet convolution – GeeksforGeeks

By Ann Roberts On Jan 6, 2023

Dirichlet convolution is a mathematical operation that combines two arithmetic functions to create a third function. It is named after the mathematician Peter Gustav Lejeune Dirichlet and is closely related to the concept of convolution in signal processing.

In mathematics, an arithmetic function is a function that takes positive integers as input and returns a number as output. The most well-known example of an arithmetic function is the divisor function, which counts the number of positive divisors a given number has.

The Dirichlet convolution of two arithmetic functions f and g is denoted by (f * g) and is defined as:

(f * g)(n) = ∑f(d) * g(n/d)

where the sum is taken over all positive divisors d of n.

Examples:

One example of a Dirichlet convolution is the convolution of the divisor function with itself, denoted by (φ * φ). This results in the Euler totient function, which is defined as:

φ(n) = ∑φ(d) * φ(n/d)

where φ(n) is the number of positive integers less than n that are relatively prime to n.

Another example of a Dirichlet convolution is the convolution of the divisor function with the function that returns 1 for all input values. This results in the identity function, which is defined as:

id(n) = ∑φ(d) * 1(n/d)

where id(n) is the function that returns n for all input values.

Dirichlet convolution can be used to calculate various arithmetic functions, including the number of divisors and the sum of a given number’s divisors.
It can also be used to find the inverse of an arithmetic function, which is a function that undoes the operation of the original function.

Approaches:

There are several approaches to implementing Dirichlet convolution, depending on the specific requirements of the application.

One simple approach is to use a brute-force method, which involves iterating over all positive divisors of the input value and summing the results of the arithmetic functions at each divisor. This approach has a time complexity of O(n), where n is the input value.

A more efficient approach is to use a sieve algorithm, which pre-calculates the values of the arithmetic functions for all integers up to a certain maximum value and stores them in an array. The Dirichlet convolution can then be calculated by simply multiplying and summing the values in the array. This approach has a time complexity of O(max), where max is the maximum value for which the arithmetic functions are pre-calculated.

Implementation of the Dirichlet convolution in Python:

Python3

def divisor_function(n):

result = 0

for d in range(1, n + 1):

if n % d == 0:

result += 1

return result

def dirichlet_convolution(f, g, n):

result = 0

for d in range(1, n + 1):

if n % d == 0:

result += f(d) * g(n//d)

return result

def gcd(a, b):

while b != 0:

a, b = b, a % b

return a

def euler_totient(n):

result = 0

for i in range(1, n + 1):

if gcd(i, n) == 1:

result += 1

return result

for i in range(1, 11):

print(f"φ({i}) = {euler_totient(i)}")

Output

φ(1) = 1
φ(2) = 1
φ(3) = 2
φ(4) = 2
φ(5) = 4
φ(6) = 2
φ(7) = 6
φ(8) = 4
φ(9) = 6
φ(10) = 4

Complexity analysis:

Time Complexity: In terms of complexity analysis, the time complexity of Dirichlet convolution depends on the specific implementation used. As mentioned earlier, the brute-force method has a time complexity of O(n), while the sieve algorithm has a time complexity of O(max).

Auxiliary Space: The space complexity of both approaches is O(n), as the values of the arithmetic functions must be stored for all positive integers up to the input value.

Optimization technique:
An optimization technique that can be used to improve the performance of Dirichlet convolution is memoization, which involves storing the results of the arithmetic functions in a cache so that they can be quickly retrieved when needed. This can greatly reduce the time required to perform the convolution, especially for large input values.

In mathematics, an arithmetic function is a function that takes positive integers as input and returns a number as output. The most well-known example of an arithmetic function is the divisor function, which counts the number of positive divisors a given number has.

The Dirichlet convolution of two arithmetic functions f and g is denoted by (f * g) and is defined as:

(f * g)(n) = ∑f(d) * g(n/d)

where the sum is taken over all positive divisors d of n.

Examples:

One example of a Dirichlet convolution is the convolution of the divisor function with itself, denoted by (φ * φ). This results in the Euler totient function, which is defined as:

φ(n) = ∑φ(d) * φ(n/d)

where φ(n) is the number of positive integers less than n that are relatively prime to n.

id(n) = ∑φ(d) * 1(n/d)

where id(n) is the function that returns n for all input values.

Dirichlet convolution can be used to calculate various arithmetic functions, including the number of divisors and the sum of a given number’s divisors.
It can also be used to find the inverse of an arithmetic function, which is a function that undoes the operation of the original function.

Approaches:

There are several approaches to implementing Dirichlet convolution, depending on the specific requirements of the application.

One simple approach is to use a brute-force method, which involves iterating over all positive divisors of the input value and summing the results of the arithmetic functions at each divisor. This approach has a time complexity of O(n), where n is the input value.

Implementation of the Dirichlet convolution in Python:

Python3

def divisor_function(n):

result = 0

for d in range(1, n + 1):

if n % d == 0:

result += 1

return result

def dirichlet_convolution(f, g, n):

result = 0

for d in range(1, n + 1):

if n % d == 0:

result += f(d) * g(n//d)

return result

def gcd(a, b):

while b != 0:

a, b = b, a % b

return a

def euler_totient(n):

result = 0

for i in range(1, n + 1):

if gcd(i, n) == 1:

result += 1

return result

for i in range(1, 11):

print(f"φ({i}) = {euler_totient(i)}")

Output

φ(1) = 1
φ(2) = 1
φ(3) = 2
φ(4) = 2
φ(5) = 4
φ(6) = 2
φ(7) = 6
φ(8) = 4
φ(9) = 6
φ(10) = 4

Complexity analysis:

Auxiliary Space: The space complexity of both approaches is O(n), as the values of the arithmetic functions must be stored for all positive integers up to the input value.

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